Much as it pains me to have to disagree with cartoon bears (or whatever they are) who make farting noises whenever Keynes’s name is uttered, I do think they have rather missed the point.
The thesis of the Keynesian multiplier, roughly speaking, is that if there are currently “idle” resources, an increase in expenditure will bring them into action, thus leading to a further increase in total income, to an extent determined by the marginal propensity to consume. This is a silly way to think about economics, for a number of reasons, but it is not internally inconsistent.
There is nothing illegal about the order of operations used from a purely mathematical standpoint – it depends on the economic assumptions involved. It seems to me that the error made by the farting bears is to stipulate at the outset that the increase in total income is limited to the amount of the original stimulus. If this were true, then yes, the MPC would have to change accordingly, and the Keynesian approach would indeed involve a mathematical sleight-of-hand. But this assumption is what Keynesians explicitly deny.
Of course, I don’t presume to speak with any authority here, so I look forward to being corrected by Professor Manish if I have missed anything.
You are correct in your criticisms of the video. The heart of the multiplier story, as you correctly point out, is that an initial “push” to the economy from either investment or government expenditure draws in previously idle resources into economic activity and thereby raises output or real income by a multiplier, the size of which is determined by the marginal propensity to consume.
So the video starts off with an incorrect premise, largely because it ignores the interaction between an initial increase in expenditure via stimulus and how that has a snowballing effect on income via the consumption function.